Mathematics

Understanding Decimals and Fractions



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A fraction is a number that can be expressed as the quotient of two integers.  A decimal represents a fraction using a period followed by tenths, hundredths, thousandths, etc.  Examples are .3=3/10, .25=25/100, .752=752/1000, and .3079=3079/10000.  A decimal number can include an integer.  Examples are 7.98=7 98/100, 12.75=12 75/100, and 100.3027=100 3027/10000.  An integer without a decimal can be, for example, any of 7, 7.0, 7.00, etc.   


A decimal can be reduced to its lowest terms when converted to a fraction.  This is done using the prime factorization of the numbers.  The decimal .2505 can be used as an example.  Since .2505=2505/10000 and the prime factorizations are 2505=3x5x167, 10000=2x2x2x2x5x5x5x5=2^4x5^4, the common denominator is 5.  Next, the common denominator (5) is divided out of the numerator (2505) and denominator (10000), ending up with 3x167/2^4x5^3=501/2000.  Even though 501=3x167 and 2000=2^4x5^3, the fraction cannot be reduced further because 3x167 and 2^4x5^3 do not have any integers in common.      


Finite decimals can be converted to fractions by adding together every place of the decimal.  For example, .349=3/10+4/100+9/1000=3x100/10x100+4X10/100X10+9/1000 =349/1000.  Also, .7507=7/10+5/100+0/1000+7/10000=7X1000/10X1000+5X100/100X100+0X1000/ 10X1000+7/10000=7507/10000.   


A decimal with an integer can be converted to a fraction using a special technique.  For example, 14.757575… can be converted to a fraction if the 75 infinitely repeats.  This is because any infinitely repeating decimal is a rational number, which is any number that can be expressed as the ratio of two integers.  Converting 14.75757575… is done using the number of integers in the infinitely repeating decimal.  Since .75757575… is 75 infinitely repeating, there are two integers infinitely repeating in the decimal.  Multiply 10 to the power of the number of integers in the infinitely repeating decimal times the given infinitely repeating decimal number, or 10^2x14.75757575=100x14.75757575=1475.757575... .  Subtract the given infinitely repeating decimal number from the given infinitely repeating decimal number times 10 to the power of the number of integers in the infinitely repeating decimal number.  Using the example, 1475.757575757575… minus 14.75757575…=1461.  Since 100y-y=99y, y=(100y-y)/99.  Using y=14.7575757575…, 100y-y=99y=1461.  Dividing both sides by 99 ends up with y=1461/99=14.757575757575….  It can be reduced to lowest terms using the prime factorization as before.  The prime factorization of 1461=3x487 and the prime factorization of 99=3x3x11.  The common denominator is 3.  Dividing both integers by 3 results in 487/3x11=487/33=14.7575757575….  


Hundredths are used to represent percentages.  All of something is 100 per cent, or 100/100=1.  Also half of something is 50/100=50%.  Other examples are 1/3=33/100=33.333%, 2/3=66.6/100=66.7%, and 4/5=80/100=80.0%.   Twice something is 2x100%= 200%, three times something is 3x100%=300%, ten times something is 10x100%= 1000%.     


Decimals can be added, subtracted, multiplied, and divided just like fractions.  Adding and subtracting decimals can be done by lining up the decimals to be added or subtracted, then subtract as usual.  Examples are .556+.411=.967 and .893-.765=.128.  In fractional form, 556/1000+411/1000=893/1000, and 893/1000-765/1000=128/1000.  Decimals are multiplied by multiplying as if integers, then moving the decimal point in the product the number of places to the left in both of the integers being multiplied.  For example, multiplying .333x.111 is done by first multiplying 333x111=36963.  Since there are three places in both the integers being multiplied, move the decimal point in the product of the two integers 3+3=6 places to the left.  The answer is .036963.    


Dividing is a little harder. Two decimals can be divided by changing the decimal in the denominator to a fraction, inverting (reversing the denominator and numerator) that fraction, then multiplying by the numerator. For example, since .333/.11 has 11/100 in the denominator, the inverted fraction to be used is 100/11. The inverted fraction is found by switching the numerator (11) and the denominator (100). The fraction 100/11 is multiplied by .333 by multiplying by the numerator (100), then dividing by the denominator (11).  The result is now 33.3/11=3.0272727272… .


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